Speed: Dimensional Formula Derivation

Speed is defined as the rate at which an object covers distance. Mathematically, it's distance travelled divided by the time taken.

Understanding Dimensions:

• Distance is a measure of length, so its dimension is Length: \([L]\).
• Time is a fundamental quantity, so its dimension is Time: \([T]\).
• Mass is not directly involved in the definition of speed, so its dimension is Mass: \([M^0]\).

Deriving the Formula

The dimensional formula for speed can be derived using its definition:

\(\text{Speed} = \frac{\text{Distance}}{\text{Time}}\)

Substituting the dimensions:

\([\text{Speed}] = \frac{[L]}{[T]}\)

Using exponent rules, we move \([T]\) to the numerator:

\([\text{Speed}] = [L^1 T^{-1}]\)

Including the dimension for mass (\([M^0]\)), the complete dimensional formula for speed is:

\([\text{Speed}] = [M^0 L^1 T^{-1}]\)

This is often written simply as \([M^0LT^{-1}]\).

Matching with Options

Comparing our derived formula \([M^0LT^{-1}]\) with the given options:

• Option 1: \([ML^0T^{-1}]\)
• Option 2: \([MLT^{-1}]\)
• Option 3: \([M^0LT^{-1}]\)
• Option 4: \([MLT^{-2}]\)

The derived dimensional formula matches Option 3.